Higgs boson production via gluon-gluon fusion and vector-boson fusion in proton-proton collisions is measured in the $H\rightarrow W W^* \rightarrow e\nu \mu\nu$ decay channel. The Large Hadron Collider delivered proton-proton collisions at a center-of-mass energy of 13 TeV between 2015 and 2018, which were recorded by the ATLAS detector, corresponding to an integrated luminosity of 139 fb$^{-1}$. The total cross sections for Higgs boson production by gluon-gluon fusion and vector-boson fusion times the $H\rightarrow W W^*$ branching ratio are measured to be $12.0\pm1.4$ and $0.75\;^{+0.19}_{-0.16}$ pb, respectively, in agreement with the Standard Model predictions of $10.4\pm 0.6$ and $0.81\pm 0.02$ pb. Higgs boson production is further characterized through measurements of Simplified Template Cross Sections in a total of 11 kinematic fiducial regions.